NONLINEAR OPERATIONS AND THE SOLUTION OF INTEGRAL EQUATIONS1 · OF INTEGRAL EQUATIONS1 BY JON C....

TRANSACTIONS OF THE ^„__.AMERICAN MATHEMATICAL SOCIETYVolume 237, March 1978

NONLINEAR OPERATIONS AND THE SOLUTIONOF INTEGRAL EQUATIONS1

BY

JON C. HELTON

Abstract. The letters S, G and H denote a linearly ordered set, a normed

complete Abelian group with zero element 0, and the set of functions from

G to G that map 0 into 0, respectively. In addition, if V G H and there

exists an additive function a from S X S to the nonnegative numbers such

that \\V'x,y)P - V'x,y)Q\\ < a(x,y)\\P - Q\\ tot each {x,y, P, Q) inS X S X G X G, then V e 6S only if fyxVP exists for each {x,y, P} in

S X S X G, and V e 6 9 only if xW(l + V)P existe for each \x,y, P) inS X S X G. It is established that V e S S if, and only if, P e 6 9. Then,this relationship is used in the solution of integral equations of the form

/(*) = h(x) + Sxc[U<u, v)f(u) + V'u, c)/(c)], where l/and Kare in SS.

This research extends known results in that requirements pertaining to the

additivity of U and V are weakened.

1. Introduction. Product integration is used to obtain solutions for linear

and nonlinear integral equations of the form

(1) f(x) = h(x)+fCV(u,v)f(v).Jx

In the linear setting, / and h are functions from S to N and F is a function

from S X S to N, where S denotes a linearly ordered set and N denotes a

normed complete ring. In the nonlinear setting, / and h are functions from S

to G and F is a function from S X S to the set of functions mapping G into

G, where G denotes a normed complete Abelian group.

We first outline a sequence of results on the solution of the linear form of

the integral equation in (1). J. S. Mac Nerney [10] introduces classes 0 & and

© 9H of functions from S X S to N such that V £ 0 & if, and only if, V is

additive and has bounded variation, and W E 0 9H if, and only if, W is

multiplicative and W — 1 has bounded variation. He then establishes a

one-to-one correspondence between the elements of 0 & and 0 9H and uses

this correspondence to obtain solutions for integral equations of the form

given in (1), where V is assumed to be an element of 0 £E;

Received by the editors June 22, 1976 and, in revised form, October 1, 1976.

AMS (MOS) subject classifications (1970). Primary 45N05; Secondary 47H15.

Key words and phrases. Sum integral, product integral, subdivision-refinement integral,nonlinear integral equation.

'This research was supported in part by a grant from Arizona State University.

© American Mathematical Society 1978

373

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

374 J. C. HELTON

The preceding results by Mac Nerney are extended by B. W. Helton [2]. He

introduces classes O A0 and OM° of functions from S X S to N such that

V E OA° if, and only if, fbaV exists and fba\V- fV\ exists and is equal to

zero for each {a, b) in S X S, and V G OM° if, and only if, 0II*(1 + V)

exists and /*|1 + V — 11(1 + V)\ exists and is equal to zero for each {a, b) in

S X S. In this treatment, the existence of

(2) jb\v-fv\=0 and J*|l + V- U(l + V)\ = 0

is used to replace the additive and multiplicative conditions required by

Mac Nerney. Helton establishes that, if V has bounded variation, then

V E O A0 if, and only if, V E OM°, and uses this correspondence to obtain

solutions for integral equations of the form

(3) f(x) = A(x) + f[ U(u, v)f(u) + V(u, v)f(v)],Jc

where U and V are assumed to be elements of O A0 having bounded variation.

These results are extensions of the results by Mac Nerney since 0ÉE is a

proper subset of the set to which V belongs if, and only if, V has bounded

variation and is in O A0.

In turn, the results by B. W. Helton are extended by J. C. Helton [4]-[6].

Here, requirements pertaining to the existence of the integrals in (2) are

dropped. It is established that, if F has bounded variation, then /* V exists for

each {a, b) in S X S if, and only if, 0II6(1 + V) exists for each (a,b) in

S x S, and this correspondence is used to obtain solutions for integral

equations of the form given in (3), where U and V are assumed to be

integrable functions of bounded variation. These results are extensions of the

results by B. W. Helton since the set of all functions in OA° with bounded

variation is a proper subset of the set to which F belongs if, and only if, F has

bounded variation and ¡baV exists for each {a,b) in S X S. However, it

should be noted that S is assumed to be a closed number interval in this

development.

We have just outlined a sequence of results on the solution of the linear

form of the integral equation in (1). We now provide a similar outline for its

nonlinear form. J. S. Mac Nerney [11] generalizes his results on the solution

of the linear equation to corresponding results for the nonlinear equation. To

do this, he redefines 0 & and 0 9H. Now, 0 & and 0 9H are sets of functions

from S X S to the set of functions mapping a normed complete Abelian

group G into itself (and 0 and into 0) such that V E 0 & if, and only if, V is

additive and there exists an additive function a from S X S to the nonnega-

tive numbers such that, if {x,y, P, Q) is in S X S X G X G, then


NONLINEAR OPERATIONS AND INTEGRAL EQUATIONS 375

(4) \\V(x,y)P - V(x,y)Q\\< a(x,y)\\P - Q\\,

and W E 0 911 if, and only if, W is multiplicative and there exists a

multiplicative function p from S X S to the set of numbers not less than one

such that, if {x,y, P, Q) is in S X S X G X G, then

\\[W(x,y)-l]P-[W(x,y)-l]Q\\<[p(x,y)-l]\\P-Q\\.

He then establishes a one-to-one correspondence between the elements of 0 &

and 0911 and uses this correspondence to obtain solutions for integral

equations fo the form given in (1), where V is assumed to be an element of

the redefined set 0 #. (Remark. Unless noted otherwise, any future reference

to 0 & or 0 911 will pertain to the sets defined in this paragraph.)

The preceding nonlinear results by J. S. Mac Nerney, as well as the linear

results by B. W. Helton, are extended by A. J. Kay [9]. He defines an analog

of the integrals in (2) for the nonlinear setting studied by Mac Nerney. In

particular, two functions M and TV from S X S to the set of functions

mapping G into G are defined to be differentially equivalent if, and only if,

there exists a function k from S x S to the nonnegative numbers such that, if

{x,y, P) is in S X S X G, then ßk = 0 and

\\M(x,y)P - N(x,y)P\\< k(x,y)\\P\\.

Kay establishes that, if V is an element of 0 6E and W is the corresponding

element of 0 911, then the set of all functions differentially equivalent to V is

the same as the set of all functions differentially equivalent to W — 1, and

uses this correspondence to obtain solutions for integral equations of the form

given in (3), where U and V are assumed to be functions which are

differentially equivalent to elements of 0 &. These results extend J. S.

Mac Nerney's results since 0 6E is a proper subset of the set of all functions

differentially equivalent to 0 &. Further, the results by B. W. Helton are also

extended since the set to which V belongs if, and only if, V has bounded

variation and F is in OA°, is a proper subset of the set of all functions

differentially equivalent to 0 &.

The purpose of this paper is to extend the results of A. J. Kay by

considering a broader class of functions than those differentially equivalent to

0 &. This extension is analogous to the extension by J. G Helton for results

obtained by B. W. Helton, where requirements involving the integrals in (2)

are dropped. In particular, it is established that, if Fis a function from S X S

to the set of all functions mapping G into G and there exists a function a

from S X S to the nonnegative numbers such that, if [x,y, P, Q) is in

S X S X G X G, then the inequality in (4) is satisfied, then /* VP exists for

each {a, b, P) in S x S x G if, and only if, an*(l + V)P exists for each

{a, b, P} in S x S X G. This correspondence is used to obtain solutions for

integral equations of the form given in (3), where U and V are assumed to be


376 J. C. HELTON

functions such that, if {x,y, P, Q) is in S X S X G X G, then each of fyxUP

and yx VP exists and each of U and V satisfies the Lipschitz condition implied

by the inequality in (4). These results extend Kay's results since the set of all

functions differentially equivalent to 0 & is a proper subset of the set of all

functions satisfying the conditions described at the end of the preceding

sentence.

In the foregoing, we have described where the results of this paper stand in

regard to other closely related papers on the solution of linear and nonlinear

forms of the integral equations in (1) and (3). These relationships are

summarized in Table 1.

Table 1. Solutions of/(x) = A(x) + (*[ U(u, v)f(u) + V(u, o)/(o)].2

Linear Results

J.S.Mac Nerney [10]

i

B. W. Helton [2]

iJ. C. Helton [4], [5], [6]

Nonlinear Results

J.S. Mac Nerney [11]

IA. J. Kay [9]

ipresent work

2. Definitions. In the following, S denotes a nondegenerate set with a linear

ordering 0, {G, +, || • ||} denotes a normed complete Abelian group with

zero element 0 and H denotes the set of all functions from G to G to which

{0, 0} belongs, with identity function 1.

If F is a function from S X S to G and {a, b) is in S X S, then the

statement that /* V exists means there exists an element L of G such that, if

e > 0, then there exists a subdivision D of {a, b) such that, if {x,}"_0 is a

refinement of D, then

/-l<E,

where V, denotes F(x,_„ x() for / = 1, 2,...,«. If V is a function from

S X S to H and {a,b,P) is in S X S X G, then the statement that

aLI6(l + V)P exists means there exists an element L of G such that, if e > 0,

then there exists a subdivision D of {a, b) such that, if {x,}",^ is a refinement

of D, then

L - fl (1 + Vt)P < e,

2Only results which pertain to the present study are included in this table.



where II"_|(1 + V¡)P denotes the image of F under continued function

composition.

If F is a function from S X S to 77, then the statement that V has

Lipschitz function a means that a is an additive function from S X S to the

nonnegative numbers such that, if [x,y, P, Q) is in S X S X G X G, then

\\V(x,y)P- V(x,y)Q\\< a(x,y)\\P - Q\\.

The statement that V satisfies the Lipschitz condition means that such a

function a exists.

The symbol 0 S denotes the set of functions from S X S to H such that V

is in 0 S if, and only if, (i) /* VP exists for each (a, b, P) in S x S X G, and

(ii) V satisfies the Lipschitz condition. The symbol 0 ^P denotes the set of

functions from S X S to H such that V is in 0 9 if, and only if, (i)

flH*(l + V)P exists for each {a, b, P) in S X S X G, and (ii) V satisfies the

Lipschitz condition. The representations 0 § and 0 $ are selected to suggest

sum integrable functions and product integrable functions, respectively.

The set of all functions differentially equivalent to Mac Nerney's set 0 &

(for the nonlinear setting) is a proper subset of 0 S. With regard to this, the

reader is referred to results by W. D. L. Appling [1, Theorem 2, p. 155] and J.

G Helton [3, p. 153].

A function h from S to G has bounded variation if, and only if, there is an

additive function ß from S X S to the nonnegative numbers such that, if

{x,y) is in S X S, then

\\h{y)-h{x)\\<ß{x,y).

Further, a function h from S to G is quasi-continuous if, and only if, there

exists a sequence {//,}£., of functions of bounded variation from 5 to G such

that, if [x,y) is in S X S, then {h¡)°°,x converges uniformly to h on (x,y).

Unless S has the least upper bound property, the preceding definition of

quasi-continuous may be requiring more than the existence of right and left

limits. For example, if S = R - {0} and h{x) = 1/x, then h has right andleft limits but is not quasi-continuous in the sense just defined.

Certain simplifying notations are now presented. Suppose [a, b) is in

S X S, {x,}"_0 is a subdivision of {a, b),f is a function defined on S and V

is a function defined on S X S. Then,/ = f{x¡) for i = 0, 1,..., n and, as

noted earlier, V¡ = V{x¡_x, x¡) for / = 1, 2,..., n. Further, if.{xy)j% de-

notes a subdivision of {x¡_x, x¡) for /* = 1,2.n, then fu = f{xy) for / =

1,2, ...,n and j = 0, 1, . . ., n(i), and Vy = V(x¡j_x, x¡¡) for i =

1,2,...,« and/ =1,2,..., n(i).

Additional background on the results in this paper and on product

integration in general can be obtained in papers by J. G Helton [6], J. G

Helton and S. Stuckwisch [7], A. J. Kay [9], B. W. Helton [2], J. S.


378 J. C. HELTON

Mac Nerney [10], [11] and P. R. Masani [12].

3. Results. We initially establish an existence theorem for sum integrals.

This result is useful later in our development.

Theorem 1. If fand g are quasi-continuous functions from S to G and V is in

0 §, then

(i) /* V(u, v)[f(u) + g(v)]and

(ii)/*[/°un[/(«) + g(«)]exist and are equal for each {a, b) in S X S.

Proof. The existence of the integral in (ii) follows by an argument similar

to the one used by J. S. Mac Nerney [11, Lemma 2.2, p. 629] to establish the

existence of right integrals of the form JbV(u, v)g(v), where V is additive

and g is quasi-continuous. We now establish that the integral in (i) exists and

is equal to the integral in (ii). Suppose {a, b) is in S X S and e > 0.

It follows from the existence of the integral in (ii) that there exists a

subdivision Dx of {a, b) such that, if {i,)7_o *s a refinement of Dx, then

¡fir y][m+m] - ,?, UvP'-<+gi <t •where f¡V = f\_V for i = 1, 2,..., m.

Let a denote a Lipschitz function for V. Since/and g are quasi-continuous,

there exists a subdivision D2 of {a, b) such that, if {s,}7-o is a refinement of

D2 and {^JJi'o is a subdivision of {s¡_,, s¡) for i = 1,2,..., m, then

m ">('')

2 2 BU-, -/«-,) + (&-*)h<*A

Let {í,}7_o denote the subdivision Z), u D2 of {a, A}. For i = 1,2,..., m,

there exists a subdivision £} of {s¡_x, s¡) such that, if {s,y}™i'0 is a refinement

of E¡, then

II /• M(f) II

¡•Vi 7-1 J"1

Let £> denote the subdivision U ^lxE¡ of (a, A}. Further, suppose {xy}"_0

is a refinement of D, {^JJl'o is a refinement of £,• for / = \,2,...,m, and

{x,};_0 is equal to U 7-Úa)j-% Now»



r[ />][/(»)+g{v)] - ¿ Vj[fj_x+gJ]\7 = 1

[HU-. + ¿r,]-2^U-1 + gJ+^3J¡ J 7=1

21 = 1

2< = 1 Lff [ft-i + g,] -2

1 = 1

m{l)

7=1[/l-l+ft]

2/=i

m(i)

2^-7 = 1

m(i)

[/-,+g,]-S 2^f[/«-l+*]1=1 y-1

+ e/3

<i=i

«(0

2Vy7 = 1

m m(i')

[/,-,+*,] -2 2 Vuífij-i+gg]1=1 y-i

+ m{e/3m) + e/3

m ">{•)

<2 2I[/*-i + ft]-[/v-i + *v]K + 2e/31=1 7=1

< e/3 + 2e/3 = e.

Therefore, the integral in (i) exists and is equal to the integral in (ii). This

completes the proof of Theorem 1.

We now establish that, if V is in 0 S, then F is in 0 9\ Four lemmas are

needed.

Lemma 2.1. 7/ F is in 0S and U{x,y)P = fyxVP for each {x,y,P) inS X S X G, then Uis in 0<éP.

Proof. This lemma follows from a result established by J. S. Mac Nerney[11, Theorem 1.1, p. 624].

Lemma 2.2. If V is in 0 S and {F„}"_0 is a sequence of elements of H suchthat

L0{x,y)P = P

and

Ln{x,y)P = P +fj(f°V))Ln_x{v,y)P

for n= 1,2,... and each {x,y, P) in S X S X G, then


380 J. C. HELTON

xIiy (l+fvy= lim L„(x,y)P

for each {x,y, P) in S X S X G.

Proof. This lemma is an adaptation of a result established by J. S.

Mac Nerney [11, Corollary 2.2, p. 630].

Lemma 2.3. If V is in 0 S and {Z,„}"„0 is a sequence of elements of H such

that

L0(x,y)P = P

and

Ln(x,y)P = P + fyV(u, v)Ln_x(v,y)PJx

for n = 1,2,... and each {x,y, P) in S X S X G, then

tn^(l+/F)F=JiirnLn(x,y)F

for each {x,y, P) in S X S X G.

Proof. This lemma follows as a corollary to Lemma 2.2 and Theorem 1

since

Xl/„v)g(ü)=fK("'r)g(ü)whenever g is a quasi-continuous function from S to G defined on {x,y).

Lemma 2.4. If V is in 0 S with Lipschitz function a and {a, b) is in S X S,

then there exists a number B such that, if w is a positive integer, e > 0 and P is

in G, then there exists a subdivision D of {a, b] such that, if {x,}7-o 's a

refinement of D, 1 < k < m, and {yj}%t¡ is a refinement of {x,}7-a» then

Jl il+ Vj)P-Lw{xk,b)P <73||FK(x„A) + e,

where L^ is defined in the statement of Lemma 2.3, ax(x,y) = a(x,y)for each

subdivision {a, x,y, b) of {a, b) and

aiix,y) =fya(u,v)a¡_x(v,y)

for i = 2, 3,... and each subdivision {a, x,y, b] of {a, b).

Proof. Let B represent a number such that, if {a, x, b) is a subdivision of

{a, A} and P is in G, then


NONLINEAR OPERATIONS AND INTEGRAL EQUATIONS

II

381

11(1+ Vt)P-Pi=l

< w\\for each subdivision {x(}"=0 of [x, b).

The lemma is established by induction on w. Suppose w = 1, e > 0 and F

is in G. There exists a subdivision D of {a, b) such that, if {x,}7=o IS a

refinement of D, 1 < k < m, and {t,}"„0 1S a refinement of {*,}7=*> then

/ VP- 2 ^ <e-■** 7=1

Suppose {x,}7L0 is a refinement of 7), 1 < k < m, and {^}"_o is a refine-

ment of {x,}7=*. Now,

n(i + ̂ )^-^.(^^7=1

7=1 9=7 + 1 J I J*k )

< 2 Vj II (1+F9)F-2^F HII 7=1 9=7 + 1 7=1

<2 fi (1 + Vq)P - pL + E7 = 1 || 9=7 + 1

< 7J||F|| 2 a, + ey-i

Thus, the lemma is established for w = 1.

The lemma is now assumed to be true for w and established for w + I.

Suppose £ > 0 and F is in G. There exists a subdivision Dx of {a, b) such

that, if {x,}7_o is a refinement of 7)„ 1 < k < m, and {>»7}"«o *s a refinement

of{*,}7=*>then

f"v{u, v^iv, b)P - 2 Vj^yj, b)P7-1

From the induction hypothesis, there exists a subdivision D2of (a,b) such

that, if {^,}7=o1S a refinement of 7>2, 1 < A: < m, and {^}"_o is a refinement

of i>,}7_„ then


382 J. C. HELTON

][ il+Vj)P-Lw{xk,b)PJ-\

< BWPW^x,, b) + e„

where e, = e[2 + 2a(a, A)]-1.

Let D denote the subdivision Dx u D2 of {a, b). Suppose {x,}7=o is a

refinement of D, 1 < k < m, and {y,}J„0 is a refinement of {x¡)"„k. Now,

í(i+r;.)p-u^^/-l

-If^+i^ n o + n)?] -Íp+/V(«,0)^(0,^111 7=1 9-y+i J * **

<|| 2 »S fi 0 + vq)p - 2 Wjj, 6)Â+b/27=1 9-7+1 7-1

<27 = 1

ft (l+KjP-Z^ty»9-7 + 1

<2[5||F|My,,A) + ei]a,. + f7-1

= 5||F||¿«,aw(y,,A) + e1¿a, + f7-1

rb

<B\\P\\J a(u,v)aw(v,b) + ^ +

= JB||F||aw+1(x„A) + C.

r2

Thus, the lemma is established for w + 1. This completes the proof of Lemma

2.4.

Theorem 2. If V is in 0 S, then V is in 0 *P and

0nV + F)p=anfc(i+jv)p

for each {a, b,P)inS X S X G.

Proof. Since V has a Lipschitz function a, it is only necessary to establish

that the product integral associated with V exists. Let {a, b, P) belong to

S X S X G. From Lemma 2.1, we have that JI6(1 + }V)P exists. In the

following, it is established that aIT*(l + V)P exists and is equal to this

integral. Let e > 0.

Let B denote a number with the properties described in Lemma 2.4. There

exists a positive integer Nx such that, if w > Nx, then



7?||F||a>, b) < b/3.

Further, it follows from Lemma 2.3 that there exists a positive integer N2 such

that, if w > N2, then

,n*(l+/f)p-L>,6)P <-

where L„{a, b) is defined in tfce statement of Lemma 2.3.

Let w be an integer greater than Nx + N2. It follows from Lemma 2.4 that

there exists a subdivision D of [a, b) such that, if {x¡)n¡m.0 is a refinement of

D, then

<B\\P\\àw{a,b) + ^

e_

3

Lw{a,b)P-Jl{l + Vi)P/-i

Suppose {x¡}n¡=0 is a refinement of D. Now,

|Lné(i + Mp-n(i + r(.)pii \ j / ,=i

<|L(a,¿)F-fi(l + F¡.)F|| +II '"I II

< 7J||PK(fl, b) + e/3 + e/3

< e/3 + e/3 + e/3 = e.

Therefore, aII*(l + V)P exists and is equal to air*(l + \V)P. This completes

the proof of Theorem 2.

We now establish that, if V is in 0 <?, then V is in 0 §. Three lemmas are

needed.

Lemma 3.1. If V is in 0 "éP and

U{x,y)P=[xJ[y{l+ V)-1]P

for each [x, y, P) in S X S X G, then UisinQS.

Proof. This lemma follows from a result established by J. S. Mac Nerney

[11, Theorem 1.1, p. 624].

Lemma 3.2. If a is an additive function from S X S to the nonnegative

numbers, [a, b) is in S X S and e > 0, then there exists a subdivision D of

{a, b) such that, if {x¡Y¡wl0 is a refinement of D and {Xy)"% is a subdivision of

{*/-!» x¡)for i=l,2,...,n, then

2i=i

»(oIT (1 + a0) - (1 + 00

7-1

<e.


384 J. C. HELTON


Mac Nerney [10, Theorem 2.3, p. 152].

Lemma 3.3. If {V¡)n¡=x is a sequence with values in H and {a¡)"„x is a

numerical sequence such that, if {P,Q) is in G X G, then

\\V(P - V&W4 ai\\P - Q\\for i = 1,2,... ,n, then (conclusion) for each {P, Q) in G X G,

O)

and

CO

n o + v,) -1i-i

p- n o + vt) -1i-i

n (i + «,) -1i-i W-Ql

no + F,.)«-ip - i + 2^

7=1

n(l + a,.)-(l +/■-l ,1,°')n-


Mac Nerney [11, Lemma 1.1, p. 623].

Theorem 3. If V is in 0 ̂ , then V is in 0 S and

/Vp=/;[n(i + F)-i]p

for each {a, b, P) in S X S X G.

Proof. Since V has a Lipschitz function a, it is only necessary to establish

that the sum integral associated with V exists. Let {a, b, P) belong to

S X S X G. From Lemma 3.1, we have that /a[II(l + V) - 1]P exists. In

the following, it is established that /* VP exists and is equal to this integral.

Let e > 0.It follows from Lemmas 3.1 and 3.2 that there exists a subdivision E of

{a, b) such that, if {sJ}'j^0 is a refinement of E and {s^}™^ is a subdivision

of {sj_x, Sj) for/ — 1, 2,..., m, then

rb m ■ j

/[II(i + f)-i1p-2 ,.,ri(i + F)-i '<!and



27=1

«CO

II (1 + «,*) - (1 + «,)¿t=i

1111 <T

For convenience, suppose E = {^}y=o-

For/ = 1, 2,..., m, there exists a subdivision 7^- of {^_i, Sj) such that, if

{i*}™-^ is a refinement of EJ} then

«GO

,ny(l + F)F- U{l + VJk)Pk = i

<3m

Let D denote the subdivision U j"=1 7^ of {a, è}. Further, suppose {x,}"=0

is a refinement of D, {í,*}a=o is a subdivision of {&_ „ Sj) for/ = 1, 2,..., m

and {x¡Y¡=0 is equal to U ^x{sJk)mk%. Now,

jT6[II(i + F)-l]F-2^

< 27 = 1

< 27=1

<27-1

<27 = 1

»..no+ k)-i

«GO

no + ri)-i*=1

F - 2 V,F{i=i

F - 2 V,P\1=1

11(1 +^-1+2 vAn

+ £ + £3^3

*=i

«GO

n (i + «,*, -

*=i«GO

1 + 2 «y*7 = 1

+ f

H+T

[Lemma 3.3]

< e/3 + 2c/3 - e.

Therefore, /_ VP exists and is equal to /a[II(l + V) - l]P. This completes

the proof of Theorem 3.

The following theorem summarizes the results of Theorems 2 and 3.

Theorem A. If V is a function from S X S to H, then the following

statements are equivalent:

{1) Vis in US,and

{2)VisinB9.


386 J. C. HELTON

Proof. This theorem follows as a corollary to Theorems 2 and 3.

The solution of the homogeneous integral equation

f{x) = P+(X[ U(u, v)f(u) + V(u, v)f(v)]

is now obtained. Four lemmas are needed.

Lemma 5.1. If b < 1, U and V are functions in 0 S with Lipschitz functions a

and ß, respectively, and ß < b, then XIV(1 — ¡V)~x(l + fU)P exists for each

{x,y,P)inS X S X G.

Proof. This lemma is an adaptation of a result established by A. J. Kay [9,

Theorem 3.2, p. 209].

Remark. For a discussion of inverses of the form (1 — F)-1, the reader is

referred to papers by J. W. Neuberger [13] and J. V. Herod [8].

Lemma 5.2. If b < 1, U and V are in 0 S with Lipschitz functions a and ß,

respectively, ß < b, {x,y, P) is in S X S X G and e > 0, then there exists a

subdivision D of {x,y) such that, if {x,}7_0 is a refinement of D and

1 < / < «, then

|i -jTf) (i + jTi/Jp - (i - r,y\i + ut)pl<e.

Proof. This lemma follows from the integrability of U and V and certain

inequalities given by J. V. Herod [8, inequalities (1) and (3), p. 188].

Lemma 5.3. If b < 1, U and V are in 0 S with Lipschitz functions a and ß,

respectively, and ß < b, then

(i)JF(l- F)-'(l+ U)Pexists and is equal to

(H)xW(l-fV)-x(l + iU)Pfor each {x,y, P) in S X S X G.

Proof. The existence of the integral in (ii) follows from Lemma 5.1. Let A'

and L denote elements of H such that

K=u+v(i- vy\i + U)

and

L-fu + fv(l-fv)~\l+fu).By applying the identity

(1) (1 - Byl(l + A) = 1 + A + 73(1 - Byx(l + A),

we have that ,1F(1 + L)P is equal to the integral in (ii). For each {x,y, P)



in S X S X G, it follows from Theorem 3 that fxLP exists and from

Theorem 2 that

(2) _n'(l+/F)p = Jl'(l + F)F.

For each [x, y, P} in S X S X G, it follows by using Theorem 1 that

and hence it follows by using Lemma 3.2 that ¡XKP exists and that

(3) fyKP= fyLP.Further, for each [x,y, P) in S X S X G, it follows by using Theorem 2

that XIF(1 + K)P exists and that

(4) tlT{l + K)P=x][y{l+JKy,

and it also follows from the identity in (1) that JF(1 + K)P is equal to the

integral in (i).

By using the equalities in (2), (3) and (4), we have

.n'o + TOP^ifo + Fypfor each [x,y, P) in S X S X G. Therefore, since JF(1 + K)P is equal to

the integral in (i) and XIF(1 + L)P is equal to the integral in (ii), the desired

equality is established. This completes the proof of Lemma 5.3.

Lemma 5.4. Ifb < 1, [c, P) is in S X G,fis a function from S to G, U and

V are functions in 0 & with Lipschitz functions a and ß, respectively, ß < b,

and U'{y, x)Q and V'{y, x)Q are equal to U{x,y)Q and V{x,y)Q, respec-

tively, for each [x,y, Q) in S X S X G, then the two statements in the

conclusion of Theorem 5 are equivalent.

Proof. This lemma is an adaptation of a result established by A. J. Kay [9,

Theorem 4.3, p. 218].

Theorem 5.1fb< 1, [c, P) is in S x G,fis a function from S to G, U and

V are functions in 0 S with Lipschitz functions a and ß, respectively, ß < b,

and U'{y, x)Q and V'{y, x)Q are equal to U{x,y)Q and V{x,y)Q, respec-

tively, for each [x,y, Q) in S X S X G, then the following statements are

equivalent:

{l)fhas bounded variation and

f{x) = P + fX[ U{u, v)f{u) + V{u, v)f{v)]Jc


388 J. C. HELTON

for each x in S, and

w fix) =,n (i - F')-'(i + u')pfor each x in S.

Proof. [(l)->(2)]. Suppose (1) is true. Then, by employing Theorem 1, we

have that

f{x) = P + fX[ U(u, v)f(u) + V(u, v)f(v)]

= P +ílííU /(«) + mfor each x in S. Now, since / U and / V satisfy the hypothesis of Lemma 5.4,

we have that

-h

/w-„n" [-(/>)']- [,♦(»'],for each x in S, where (/^F)'g and (/^i/)'Ô are equal to jvuVQ and fvuUQ,

respectively, for each {u, v, Q) in S X S X G. Thus, by applying Lemma

5.3, we have that

f(x)=xUe(\-vyXi + u')p

for each x in S. Therefore, (1) implies (2).

Proof. [(2) -> (1)]. Suppose (2) is true. Then, by applying Lemma 5.3, we

have that

/«-.IT [i-(H]"'[i + (/")>for each x in 5, where (¡V)' and (JU)' are defined as before. Thus, by

applying Lemma 5.4, we have that

™-p+n\£u\'M+\£r m

- P + H U(u, v)f(u) + V(u, v)f(v)},J r

where the second equality follows from Theorem 1. The bounded variation of

/ follows readily. Therefore, (2) implies (1). This completes the proof of

Theorem 5.

In the preceding, we have restricted our study to interval functions in H.

This restriction is now relaxed by considering a new class H* of functions

such that, if F is a function from S X S to the set of functions mapping G



into G, then V is in H* if, and only if, there exists an additive function y from

S x S to the nonnegative numbers such that, if (x,y) is in S X S, then

/_ VO exists and

\\V{x,y)0\\< y{x,y).

By introducing 77*, we are weakening the restraint in the definition of H that

functions must map 0 into 0. Let 0 S * and 0 ^P * be defined with respect to

77* as 0 § and 0 •_? are defined with respect to 77. The following theorems

follow as corollaries to Theorems 1, 4 and 5, respectively, by using a

technique presented by J. S. Mac Nerney [11, Remark, p. 637].

Theorem 6. If fand g are quasi-continuous functions from S to G and V is in

0 S *, then /_. V{u, v)[f{u) + g{v)] exists for each {a, b) in S X S.

Theorem 7. If V is a function from S X S to H*, then the following

statements are equivalent:

{1) Vis in OS*, and

{2) Vis in 69*.

Theorem 8.7/6 < 1, {c, F) is in S X G,fis a function from S to G, U andV are functions in 0 S * with Lipschitz functions a and ß, respectively, each of ß

and || K0|| is less than b, and U'{y, x)Q and V'{y, x)Q are equal to U{x,y)Q

and V{x,y)Q, respectively, for each [x,y, Q) in S X S X G, then the fol-

lowing statements are equivalent:

{l)fhas bounded variation and

f{x) = P + fX[ U{u, v)f{u) + V{u, v)f{v)]Jc


(2) f(x) =JL\l - V)-\l + U')P

for each x in S.

Now that Theorem 8 is available, it is easy to solve the nonhomogeneous

integral equation

f(x) = h{x) + H U{u, v)f{u) + V{u, v)f{v)],Jc

where A is a function of bounded variation from S to G.

Theorem 9. If b < 1, c is in S, f and h are functions from S to G, h has

bounded variation, U and V are functions in 0 S * with Lipschitz functions a and

ß, respectively, each of ß and \\V0\\ is less than b, T{x,y)Q = U{x,y)Q +

fy dh for each {x,y, Q) in S X S X G, and T'{y, x)Q and V'{y, x)Q are

equal to T{x,y)Q and V{x,y)Q, respectively, for each {x,y, Q) in S X S X


390 J. C. HELTON

G, then the following statements are equivalent:

(l)fhas bounded variation and

fix) = A(x) + H U{u, v)f(u) + V(u, v)f(v)]Jc


(2) f{x) =xU\l - Vyx(l + T')h(c)

for each x in S.

Proof. The integral equation in (1) can be rewritten as

f{x) = h(c) + ['[ T(u, v)f(u) + V(u, v)f(v)].Jc

Then the theorem follows as a corollary to Theorem 8.

Bibliography

1. W. D. L. Appling, Interval functions and real Hilbert spaces, Rend. Circ. Mat. Palermo (2)

11 (1962), 154-156. MR 27 #4040.2. B. W. Helton, Integral equations and product integrals. Pacific J. Math. 16 (1966), 297-322.

MR 32 #6167.3. J. C. Helton, An existence theorem for sum and product integrals, Proc. Amer. Math. Soc. 39

(1973), 149-154. MR 47 #5596.4._, Mutual existence of sum and product integrals. Pacific J. Math. 56 (1975), 495-516.5._, Product integrals and the solution of integral equations, Pacific J. Math. 58 (1975),

87-103. MR 52 #6341.6. _, Existence of integrals and the solution of integral equations, Trans. Amer. Math.

Soc. 229 (1977), 307-327.7. J. C. Helton and S. Stuckwisch, Numerical approximation of product integrals, J. Math.

Anal. Appl. 56 (1976), 41(M37.8. J. V. Herod, Coalescence of solutions for nonlinear Stieltjes equations, J. Reine Angew. Math.

252 (1972), 187-194. MR 45 #996.9. A. J. Kay, Nonlinear integral equations and product integrals, Pacific J. Math. 60 (1975),

203-222.10. J. S. Mac Nerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148-173.

MR 26 #1726.

11._, A nonlinear integral operation, Illinois J. Math. 8 (1964), 621-638. MR 29 #5082.12. P. R. Masani, Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc.

61 (1947), 147-192. MR 8, 321.13. J. W. Neuberger, Toward a characterization of the identity component of rings and near-rings

of continuous transformations, J. Reine Angew. Math. 238 (1969), 100-104. MR 40 #3384.

Department of Mathematics, Arizona State University, Tempe, Arizona 85281


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